Comptes Rendus
Mathematical Analysis
The zero-one law for a complete orthonormal system
Comptes Rendus. Mathématique, Volume 339 (2004) no. 5, pp. 335-337.

A complete orthonormal system of functions Θ={θn}n=1,θnL[0,1] is constructed such that n=1anθn converges almost everywhere on [0,1] if {an}n=1l2 and n=1anθn diverges a.e. for any {an}n=1l2. We also show that for any complete ONS {fn}n=1 of functions defined on [0,1] there exists a fixed non decreasing subsequence {nk}k=1 of natural numbers such that for any fL[0,1]0 and some sequence of coefficients {bn}n=1,

n=1nkbnfnfa.e. whenk.

On construit un système orthonormal complet Θ={θn}n=1,θnL[0,1] tel que n=1anθn converge presque partout pour n'importe quel {an}n=1l2 et diverge presque partout pour n'importe quel {an}n=1l2. Nous démontrons que pour toute système orthonormal complet {fn}n=1 il existe une sous suite croissante {nk}k=1 d'entiers naturels tels que pour tout fL[0,1]0 il existe une suite de coefficients tels que

n=1Nkbnfnfp.p. sik.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.07.009
Kazaros Kazarian 1

1 Departamento de Matemáticas,C-XV, Universidad Autónoma de Madrid, 28049 Madrid, Spain
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Kazaros Kazarian. The zero-one law for a complete orthonormal system. Comptes Rendus. Mathématique, Volume 339 (2004) no. 5, pp. 335-337. doi : 10.1016/j.crma.2004.07.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.07.009/

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